Simple orthogonal transforms such as the Haar transform have been used in image recovery, but communications channels tend to use techniques such as the Hadamard transform for forward error corrections. The complexity of transformation for forward error correction often requires complex adaptive and dynamic quantization techniques and increased processing times. For example, a Haar transform takes a set of discrete samples and transforms the samples into a set of real value samples in an infinite field. Even though the transform is invertible, transmitting real value samples in a digital communications channel requires a dynamically adaptive quantizer which increases processing complexity. On the other hand, if the real value samples are modified to contain only a set of finite values and create a finite field set, they are not invertible exactly and therefore will have errors in reconstruction. From a digital transmission point of view, the transformation must be invertible to recover data and allow the samples to be in a finite field to ensure that they can be transmitted using any digital modulation technique. Therefore, a Haar transform, while used in video compression techniques, has not be used in error corrections due to impairments in a digital communications channel.
Also, Wireless channels which have multi-path fading impairments and Doppler effects often make it difficult to use complex modulations. They also tend to be bandwidth limited and mobility of the nodes affect maintaining continuous path connectivity and thus causes significant performance degradations.
The proposed invention is based on a unified newly developed Koay-Vaman (KV) transform that uses “orthonormal transformation” which is easily invertible. It takes a discrete sample set and transforms it into real value samples that are contained in a finite field. Therefore, the real value samples can be coded using any digital modulation technique whereby the invertible transform property is combined with the values of each output sample in the “finite field”. This allows use of the transform for error recovery in the physical layer of the network as a form of “Forward Error Correction”. Also, the error correction can be done selectively in a finite amount of time which is critical to maintain end-to-end delay to a minimum value. This is superior to existing error correction techniques such as Automatic Repeat Request (ARQ) used at the link layer where packet retransmission can cause errors in the transmission as part of error correction and cause further delay to become random and extensive. It is also less complex and cumbersome compared to more common error correction techniques used today including “Hadamard transform” and “Reed-Solomon Coding” and “turbo-coding”.
The performance of the proposed invention is demonstrated when using Quadrature Amplitude Modulation as the digital modulation over the channel. It achieves minimization of the impact of multi-path interference (fading) and Doppler effects in the wireless channel. The performance of the overall system in the presence of multi-path fading and Doppler effects plus AWGN is compared to that of the presence of AWGN only.